一、线性回归
1、简单线性回归
a、
> x = women
> x
height weight
1 58 115
2 59 117
3 60 120
4 61 123
5 62 126
6 63 129
7 64 132
8 65 135
9 66 139
10 67 142
11 68 146
12 69 150
13 70 154
14 71 159
15 72 164
> fit = lm(weight ~ height, data=x)
> summary(fit) Call:
lm(formula = weight ~ height, data = x) Residuals:
Min 1Q Median 3Q Max
-1.7333 -1.1333 -0.3833 0.7417 3.1167 Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -87.51667 5.93694 -14.74 1.71e-09 ***
height 3.45000 0.09114 37.85 1.09e-14 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.525 on 13 degrees of freedom
Multiple R-squared: 0.991, Adjusted R-squared: 0.9903
F-statistic: 1433 on 1 and 13 DF, p-value: 1.091e-14 > fitted(fit)
1 2 3 4 5 6 7 8 9
112.5833 116.0333 119.4833 122.9333 126.3833 129.8333 133.2833 136.7333 140.1833
10 11 12 13 14 15
143.6333 147.0833 150.5333 153.9833 157.4333 160.8833
> women$weight
[1] 115 117 120 123 126 129 132 135 139 142 146 150 154 159 164
> residuals(fit)
1 2 3 4 5 6 7
2.41666667 0.96666667 0.51666667 0.06666667 -0.38333333 -0.83333333 -1.28333333
8 9 10 11 12 13 14
-1.73333333 -1.18333333 -1.63333333 -1.08333333 -0.53333333 0.01666667 1.56666667
15
3.11666667
> plot(women$height, women$weight)
> abline(fit)
b、
> x = runif(100)
> y = 0.2*x + 0.1*rnorm(100)
> fit = lm(y~x)
> summary(fit) Call:
lm(formula = y ~ x) Residuals:
Min 1Q Median 3Q Max
-0.299493 -0.056850 0.004709 0.066714 0.237272 Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.002891 0.019688 -0.147 0.884
x 0.236938 0.036158 6.553 2.64e-09 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.1037 on 98 degrees of freedom
Multiple R-squared: 0.3047, Adjusted R-squared: 0.2976
F-statistic: 42.94 on 1 and 98 DF, p-value: 2.639e-09 > plot(x,y)
> abline(fit)
c、
> y = 0.2*x + 0.01*rnorm(100)
> fit = lm(y~x)
> summary(fit) Call:
lm(formula = y ~ x) Residuals:
Min 1Q Median 3Q Max
-0.019936 -0.005549 -0.001135 0.004598 0.026435 Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.002684 0.001837 -1.461 0.147
x 0.203561 0.003374 60.326 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.009678 on 98 degrees of freedom
Multiple R-squared: 0.9738, Adjusted R-squared: 0.9735
F-statistic: 3639 on 1 and 98 DF, p-value: < 2.2e-16 > plot(x,y)
> abline(fit)
2、多项式线性回归
a、
> fit2 = lm(weight ~ height + I(height^2), data=women)
> summary(fit2) Call:
lm(formula = weight ~ height + I(height^2), data = women) Residuals:
Min 1Q Median 3Q Max
-0.50941 -0.29611 -0.00941 0.28615 0.59706 Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 261.87818 25.19677 10.393 2.36e-07 ***
height -7.34832 0.77769 -9.449 6.58e-07 ***
I(height^2) 0.08306 0.00598 13.891 9.32e-09 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.3841 on 12 degrees of freedom
Multiple R-squared: 0.9995, Adjusted R-squared: 0.9994
F-statistic: 1.139e+04 on 2 and 12 DF, p-value: < 2.2e-16 > plot(women$height, women$weight)
> lines(women$height, fitted(fit2))
b、
> y = 0.4*x**2 + 0.2*x + 0.01*rnorm(100)
> fit = lm(y~x + I(x^2))
> summary(fit) Call:
lm(formula = y ~ x + I(x^2)) Residuals:
Min 1Q Median 3Q Max
-0.0243909 -0.0058432 -0.0000949 0.0056788 0.0245737 Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.003611 0.002727 1.324 0.189
x 0.189098 0.013571 13.934 <2e-16 ***
I(x^2) 0.400631 0.013806 29.018 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.009857 on 97 degrees of freedom
Multiple R-squared: 0.9966, Adjusted R-squared: 0.9965
F-statistic: 1.418e+04 on 2 and 97 DF, p-value: < 2.2e-16 > plot(x,y)
3、多元线性回归
> states = as.data.frame(state.x77[, c("Murder", "Population", "Illiteracy", "Income", "Frost")])
> cor(states)
Murder Population Illiteracy Income Frost
Murder 1.0000000 0.3436428 0.7029752 -0.2300776 -0.5388834
Population 0.3436428 1.0000000 0.1076224 0.2082276 -0.3321525
Illiteracy 0.7029752 0.1076224 1.0000000 -0.4370752 -0.6719470
Income -0.2300776 0.2082276 -0.4370752 1.0000000 0.2262822
Frost -0.5388834 -0.3321525 -0.6719470 0.2262822 1.0000000
> install.packages("car")
> library(car)
> scatterplotMatrix(states, spread=FALSE, lty.smooth=2, main="Scatter Plot Matrix")
> fit = lm(Murder ~ Population + Illiteracy + Income + Frost, data=states)
> summary(fit) Call:
lm(formula = Murder ~ Population + Illiteracy + Income + Frost,
data = states) Residuals:
Min 1Q Median 3Q Max
-4.7960 -1.6495 -0.0811 1.4815 7.6210 Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.235e+00 3.866e+00 0.319 0.7510
Population 2.237e-04 9.052e-05 2.471 0.0173 *
Illiteracy 4.143e+00 8.744e-01 4.738 2.19e-05 ***
Income 6.442e-05 6.837e-04 0.094 0.9253
Frost 5.813e-04 1.005e-02 0.058 0.9541
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.535 on 45 degrees of freedom
Multiple R-squared: 0.567, Adjusted R-squared: 0.5285
F-statistic: 14.73 on 4 and 45 DF, p-value: 9.133e-08
4、有叫互项的多元线性回归
> fit = lm(mpg ~ hp + wt + hp:wt, data=mtcars)
> summary(fit) Call:
lm(formula = mpg ~ hp + wt + hp:wt, data = mtcars) Residuals:
Min 1Q Median 3Q Max
-3.0632 -1.6491 -0.7362 1.4211 4.5513 Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 49.80842 3.60516 13.816 5.01e-14 ***
hp -0.12010 0.02470 -4.863 4.04e-05 ***
wt -8.21662 1.26971 -6.471 5.20e-07 ***
hp:wt 0.02785 0.00742 3.753 0.000811 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.153 on 28 degrees of freedom
Multiple R-squared: 0.8848, Adjusted R-squared: 0.8724
F-statistic: 71.66 on 3 and 28 DF, p-value: 2.981e-13
马力与车重的叫互项是显著的，说明：响应变量与其中一个预测变量的关系依赖于另外一个预测变量的水平。
> install.packages("effects")
> library(effects)
> plot(effect("hp:wt", fit, list(wt=c(2.2, 3.2, 4.2))), multiline=TRUE)
